Answer:
Height = 20.6 cm
Explanation:
We can solve this problem by using the conservation of mechanical energy.
In fact, for each of the two objects, the initial gravitational potential energy when it is at the top of the ramp is entirely converted into translational kinetic energy + rotational kinetic energy. So we can write for each object
Solid Sphere
KE1 + PE1 = KE2 + PE2
[tex]0 + m g h = \frac{1}{2} m v^2 + \frac{1}{2} I w^2 + 0\\\\I = \frac{2}{5} m r^2\\\\m g h = 0.5 m v^2 + 0.5 * 0.4 m r^2 (v/r)^2\\\\m g h = 0.5 m v^2 + 0.2 m v^2[/tex]
[tex]g h = 0.7 v^2\\\\v = \sqrt{(gh/0.7)} \\\\= \sqrt{( 9.81 \times 0.193 / 0.7)} \\\\= 1.645 m/s[/tex]
Solid Cylinder
KE1 + PE1 = KE2 + PE2
[tex]0 + m g h = \frac{1}{2} m v^2 + \frac{1}{2} I w^2 + 0[/tex]
[tex]I = \frac{1}{2} m r^2[/tex]
[tex]m g h = \frac{1}{2} m v^2 + \frac{1}{2} * \frac{1}{2} m r^2 (v/r)^2[/tex]
[tex]m g h = 0.5 m v^2 + 0.25 m v^2[/tex]
[tex]g h = 0.75 v^2[/tex]
[tex]h =\frac{ 0.75 v^2}{g} \\\\=\frac{ 0.75 \times (1.645)^2}{ 9.81} \\\\= 0.206 m \\\\= 20.6 cm[/tex]
Height = 20.6 cm
